\(\int \frac {(1-x^3)^{2/3} (-1+4 x^3)}{x^6 (-2+3 x^3)} \, dx\) [2072]

Optimal result

Integrand size = 31, antiderivative size = 150 \[ \int \frac {\left (1-x^3\right )^{2/3} \left (-1+4 x^3\right )}{x^6 \left (-2+3 x^3\right )} \, dx=\frac {\left (1-x^3\right )^{2/3} \left (-4+29 x^3\right )}{40 x^5}-\frac {5 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1-x^3}}\right )}{4\ 2^{2/3} \sqrt {3}}+\frac {5 \log \left (-x+\sqrt [3]{2} \sqrt [3]{1-x^3}\right )}{12\ 2^{2/3}}-\frac {5 \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1-x^3}+2^{2/3} \left (1-x^3\right )^{2/3}\right )}{24\ 2^{2/3}} \]

[Out]

1/40*(-x^3+1)^(2/3)*(29*x^3-4)/x^5-5/24*arctan(3^(1/2)*x/(x+2*2^(1/3)*(-x^3+1)^(1/3)))*2^(1/3)*3^(1/2)+5/24*ln
(-x+2^(1/3)*(-x^3+1)^(1/3))*2^(1/3)-5/48*ln(x^2+2^(1/3)*x*(-x^3+1)^(1/3)+2^(2/3)*(-x^3+1)^(2/3))*2^(1/3)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.84, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {594, 597, 12, 384} \[ \int \frac {\left (1-x^3\right )^{2/3} \left (-1+4 x^3\right )}{x^6 \left (-2+3 x^3\right )} \, dx=-\frac {5 \arctan \left (\frac {\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{4\ 2^{2/3} \sqrt {3}}-\frac {5 \log \left (3 x^3-2\right )}{24\ 2^{2/3}}+\frac {5 \log \left (\frac {x}{\sqrt [3]{2}}-\sqrt [3]{1-x^3}\right )}{8\ 2^{2/3}}-\frac {\left (1-x^3\right )^{2/3}}{10 x^5}+\frac {29 \left (1-x^3\right )^{2/3}}{40 x^2} \]

[In]

Int[((1 - x^3)^(2/3)*(-1 + 4*x^3))/(x^6*(-2 + 3*x^3)),x]

[Out]

-1/10*(1 - x^3)^(2/3)/x^5 + (29*(1 - x^3)^(2/3))/(40*x^2) - (5*ArcTan[(1 + (2^(2/3)*x)/(1 - x^3)^(1/3))/Sqrt[3
]])/(4*2^(2/3)*Sqrt[3]) - (5*Log[-2 + 3*x^3])/(24*2^(2/3)) + (5*Log[x/2^(1/3) - (1 - x^3)^(1/3)])/(8*2^(2/3))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 594

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*g*(m + 1))), x] - Dist[1/(a*g^n*(m + 1
)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c*(p + 1) + a*d*q)
+ d*((b*e - a*f)*(m + 1) + b*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && IGtQ[n
, 0] && GtQ[q, 0] && LtQ[m, -1] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (1-x^3\right )^{2/3}}{10 x^5}-\frac {1}{10} \int \frac {-29+31 x^3}{x^3 \sqrt [3]{1-x^3} \left (-2+3 x^3\right )} \, dx \\ & = -\frac {\left (1-x^3\right )^{2/3}}{10 x^5}+\frac {29 \left (1-x^3\right )^{2/3}}{40 x^2}-\frac {1}{40} \int -\frac {50}{\sqrt [3]{1-x^3} \left (-2+3 x^3\right )} \, dx \\ & = -\frac {\left (1-x^3\right )^{2/3}}{10 x^5}+\frac {29 \left (1-x^3\right )^{2/3}}{40 x^2}+\frac {5}{4} \int \frac {1}{\sqrt [3]{1-x^3} \left (-2+3 x^3\right )} \, dx \\ & = -\frac {\left (1-x^3\right )^{2/3}}{10 x^5}+\frac {29 \left (1-x^3\right )^{2/3}}{40 x^2}-\frac {5 \arctan \left (\frac {1+\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{4\ 2^{2/3} \sqrt {3}}-\frac {5 \log \left (-2+3 x^3\right )}{24\ 2^{2/3}}+\frac {5 \log \left (\frac {x}{\sqrt [3]{2}}-\sqrt [3]{1-x^3}\right )}{8\ 2^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.89 \[ \int \frac {\left (1-x^3\right )^{2/3} \left (-1+4 x^3\right )}{x^6 \left (-2+3 x^3\right )} \, dx=\frac {1}{240} \left (-\frac {24 \left (1-x^3\right )^{2/3}}{x^5}+\frac {174 \left (1-x^3\right )^{2/3}}{x^2}-50 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2-2 x^3}}\right )+50 \sqrt [3]{2} \log \left (-x+\sqrt [3]{2-2 x^3}\right )-25 \sqrt [3]{2} \log \left (x^2+x \sqrt [3]{2-2 x^3}+\left (2-2 x^3\right )^{2/3}\right )\right ) \]

[In]

Integrate[((1 - x^3)^(2/3)*(-1 + 4*x^3))/(x^6*(-2 + 3*x^3)),x]

[Out]

((-24*(1 - x^3)^(2/3))/x^5 + (174*(1 - x^3)^(2/3))/x^2 - 50*2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(2 - 2*x
^3)^(1/3))] + 50*2^(1/3)*Log[-x + (2 - 2*x^3)^(1/3)] - 25*2^(1/3)*Log[x^2 + x*(2 - 2*x^3)^(1/3) + (2 - 2*x^3)^
(2/3)])/240

Maple [A] (verified)

Time = 13.59 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.11

method	result	size
pseudoelliptic	\(\frac {50 \sqrt {3}\, 2^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3}\, \left (x +2 \,2^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}-25 \,2^{\frac {1}{3}} x^{5} \ln \left (2\right )+50 \,2^{\frac {1}{3}} x^{5} \ln \left (\frac {-2^{\frac {2}{3}} x +2 {\left (-\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}}}{x}\right )-25 \,2^{\frac {1}{3}} x^{5} \ln \left (\frac {2^{\frac {2}{3}} {\left (-\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}} x +2^{\frac {1}{3}} x^{2}+2 {\left (-\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )+174 \left (-x^{3}+1\right )^{\frac {2}{3}} x^{3}-24 \left (-x^{3}+1\right )^{\frac {2}{3}}}{240 x^{5}}\)	\(167\)
risch	\(\text {Expression too large to display}\)	\(908\)
trager	\(\text {Expression too large to display}\)	\(1127\)

[In]

int((-x^3+1)^(2/3)*(4*x^3-1)/x^6/(3*x^3-2),x,method=_RETURNVERBOSE)

[Out]

1/240*(50*3^(1/2)*2^(1/3)*arctan(1/3*3^(1/2)/x*(x+2*2^(1/3)*(-x^3+1)^(1/3)))*x^5-25*2^(1/3)*x^5*ln(2)+50*2^(1/
3)*x^5*ln((-2^(2/3)*x+2*(-(-1+x)*(x^2+x+1))^(1/3))/x)-25*2^(1/3)*x^5*ln((2^(2/3)*(-(-1+x)*(x^2+x+1))^(1/3)*x+2
^(1/3)*x^2+2*(-(-1+x)*(x^2+x+1))^(2/3))/x^2)+174*(-x^3+1)^(2/3)*x^3-24*(-x^3+1)^(2/3))/x^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (114) = 228\).

Time = 2.02 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.85 \[ \int \frac {\left (1-x^3\right )^{2/3} \left (-1+4 x^3\right )}{x^6 \left (-2+3 x^3\right )} \, dx=\frac {100 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{5} \arctan \left (\frac {4^{\frac {1}{6}} {\left (12 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (3 \, x^{4} - 2 \, x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} \sqrt {3} {\left (27 \, x^{9} - 72 \, x^{6} + 36 \, x^{3} + 8\right )} + 12 \, \sqrt {3} {\left (9 \, x^{8} - 6 \, x^{5} - 4 \, x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (27 \, x^{9} - 36 \, x^{3} + 8\right )}}\right ) + 50 \cdot 4^{\frac {2}{3}} x^{5} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 4^{\frac {2}{3}} {\left (3 \, x^{3} - 2\right )} - 12 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x}{3 \, x^{3} - 2}\right ) - 25 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x - 4^{\frac {1}{3}} {\left (9 \, x^{6} - 6 \, x^{3} - 4\right )} - 6 \, {\left (3 \, x^{5} - 4 \, x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{9 \, x^{6} - 12 \, x^{3} + 4}\right ) + 36 \, {\left (29 \, x^{3} - 4\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{1440 \, x^{5}} \]

[In]

integrate((-x^3+1)^(2/3)*(4*x^3-1)/x^6/(3*x^3-2),x, algorithm="fricas")

[Out]

1/1440*(100*4^(1/6)*sqrt(3)*x^5*arctan(1/6*4^(1/6)*(12*4^(2/3)*sqrt(3)*(3*x^4 - 2*x)*(-x^3 + 1)^(2/3) - 4^(1/3
)*sqrt(3)*(27*x^9 - 72*x^6 + 36*x^3 + 8) + 12*sqrt(3)*(9*x^8 - 6*x^5 - 4*x^2)*(-x^3 + 1)^(1/3))/(27*x^9 - 36*x
^3 + 8)) + 50*4^(2/3)*x^5*log(-(6*4^(1/3)*(-x^3 + 1)^(1/3)*x^2 - 4^(2/3)*(3*x^3 - 2) - 12*(-x^3 + 1)^(2/3)*x)/
(3*x^3 - 2)) - 25*4^(2/3)*x^5*log((6*4^(2/3)*(-x^3 + 1)^(2/3)*x - 4^(1/3)*(9*x^6 - 6*x^3 - 4) - 6*(3*x^5 - 4*x
^2)*(-x^3 + 1)^(1/3))/(9*x^6 - 12*x^3 + 4)) + 36*(29*x^3 - 4)*(-x^3 + 1)^(2/3))/x^5

Sympy [F]

\[ \int \frac {\left (1-x^3\right )^{2/3} \left (-1+4 x^3\right )}{x^6 \left (-2+3 x^3\right )} \, dx=\int \frac {\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \cdot \left (4 x^{3} - 1\right )}{x^{6} \cdot \left (3 x^{3} - 2\right )}\, dx \]

[In]

integrate((-x**3+1)**(2/3)*(4*x**3-1)/x**6/(3*x**3-2),x)

[Out]

Integral((-(x - 1)*(x**2 + x + 1))**(2/3)*(4*x**3 - 1)/(x**6*(3*x**3 - 2)), x)

Maxima [F]

\[ \int \frac {\left (1-x^3\right )^{2/3} \left (-1+4 x^3\right )}{x^6 \left (-2+3 x^3\right )} \, dx=\int { \frac {{\left (4 \, x^{3} - 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (3 \, x^{3} - 2\right )} x^{6}} \,d x } \]

[In]

integrate((-x^3+1)^(2/3)*(4*x^3-1)/x^6/(3*x^3-2),x, algorithm="maxima")

[Out]

integrate((4*x^3 - 1)*(-x^3 + 1)^(2/3)/((3*x^3 - 2)*x^6), x)

Giac [F]

\[ \int \frac {\left (1-x^3\right )^{2/3} \left (-1+4 x^3\right )}{x^6 \left (-2+3 x^3\right )} \, dx=\int { \frac {{\left (4 \, x^{3} - 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (3 \, x^{3} - 2\right )} x^{6}} \,d x } \]

[In]

integrate((-x^3+1)^(2/3)*(4*x^3-1)/x^6/(3*x^3-2),x, algorithm="giac")

[Out]

integrate((4*x^3 - 1)*(-x^3 + 1)^(2/3)/((3*x^3 - 2)*x^6), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1-x^3\right )^{2/3} \left (-1+4 x^3\right )}{x^6 \left (-2+3 x^3\right )} \, dx=\int \frac {{\left (1-x^3\right )}^{2/3}\,\left (4\,x^3-1\right )}{x^6\,\left (3\,x^3-2\right )} \,d x \]

[In]

int(((1 - x^3)^(2/3)*(4*x^3 - 1))/(x^6*(3*x^3 - 2)),x)

[Out]

int(((1 - x^3)^(2/3)*(4*x^3 - 1))/(x^6*(3*x^3 - 2)), x)